# Traversing a circular arc

## Homework Statement

A car is tested along specifically designed track. First, the car is driven along a straight section of track of length (r). If the car starts from rest under a maximum (constant) acceleration, it takes an amount of time (t) to cover the distance (r). The car is then brought to rest, and is continually accelerated along a section of track that makes a circular arc of radius (r). Assuming that the car is speeding up at the maximum possible rate that allows it to remain on the track, how long would it take to traverse the arc?

I don't know how to write equation so I am sub-scripting my variables by attaching _* at the end.

## Homework Equations

V_avg = distance / time
V_avg = (V_f + V_o) / 2
acceleration = (V_f - V_o) / time
a_centripetal = v^2 / r

## The Attempt at a Solution

V_o = 0
a = V_f / time
V_avg = V_f / 2
V_f = 2 * V_avg
V_avg = distance / time = r/t
V_f = 2 * (r/t)
a = 2 * r / t^2
a_centripetal = a (because the car is speeding up at the maximum rate, is this logic correct??)
so...
a_centripetal = (2 * r) / t^2
but a_centripetal = v^2 / r
so v^2 / r = (2 * r) / t^2
solving for v:
v = r/t * sqrt(2)

This is where I am stuck, I found the tangential velocity, but what do I do next?

Say the arc length is created by an angle theta, but theta isn't given, but the arc length would be theta * r, but where would I go next if I go with this route?

Or I could somehow relate the tangential velocity with the angular velocity... but how...??? And I would still need theta right?

I asked my professor and he said that the problem is stated correctly and I don't need to know the coefficient of friction or the arc length.. so I am totally lost >_>

EDIT: I remember posting this problem like a day or two ago but I can't seem to find that thread Can anyone make a suggestion on how I can proceed with this problem?

Thank you!!!

Should I be using the equation time = 2 * pi * r / velocity?

But the arc is not a full circle?

I need an insight, I am staying up all night but still can't figure this one out. It's due in like 5 hours....

gneill
Mentor
The problem is not well posed or is incomplete. While the radius of the arc is given as r, there is no information about it's length (either arc length or angular). "Circular arc" can mean anything from zero to 360° around a circle.

Also, "Assuming that the car is speeding up at the maximum possible rate that allows it to remain on the track" is meaningless without information about the friction between the car and road surface, the extent of any road cambering or banking, and so on. It implies that the acceleration of the car may be different from the straight-line case, and there's no information available to know what it might be.

Sorry I forgot to update this.

So my professor explained how to solve it and apparently question is formed correctly and does not need to be fixed or anything.

The way to solve is to observe that the maximum acceleration the car can have on the track is found by solving for the acceleration in part 1, where the track is straight.

Then the next part is to recognize that the car does not have the same velocity throughout its travel in the circular arc, it starts from 0 and ends with some final velocity.

Then you have to realize that when the car can no longer be on the track, its centripetal acceleration has to be equal to what the maximum acceleration that was found on the first part. which means a = 2 * r / t ^2 = v_final ^ 2 /r. Solving for v_final tells you the final tangential velocity of the car before it can no longer be on the track.

Then what you have to solve is how long it takes for the car to achieve that final tangential velocity, namely a_tangential * t_tangential = v_f. Then t_tangential would be the time it takes for the car to traverse the circular arc.

That is all he told me, I assumed that the a_tangential also has to be equal to the acceleration in the first part, which is a = 2 * r / t^2.

So plugging in the acceleration and the v_f in a_tangential * t_tangential = v_f and this would allow you to solve for t_tangential.

The result I got is that the time it took for the car to traverse the arc is t_tangential = (sqrt (2) / 2) * t

Can anyone verify if this is correct? I know this problem is really hard and if you are right and that this problem could not be solved without knowing the coefficient of friction or the length of the arc, please tell me because I will talk to him and maybe he won't take off the score on that problem.

Thank you!!! :) :) :)

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